King safety is a central issue in the Jerome Gambit (1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+). In fact, White sacrifices two pieces in order to render his opponent's King as unsafe as possible.
What about White's King?
It is clear that certain variations are focused upon a counter-attack by Black on White's King, for example 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Kxf7 5.Nxe5+ Nxe5 6.d4 Qh4!?
But, in general?
I recently had a short discussion about the Jerome Gambit with Stockfish 15 (a skeptic) and The Database (a supporter).
After 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.Bxf7+ Stockfish rated Black as about 2 3/4 pawns better. On the other hand, The Database, with 31,564 games with that position, showed that White scored 52%.
What happens, though, if White first gives some thought to his own King's safety before sacrificing, say by castling or developing another piece?
It turns out that after 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.0-0 Stockfish 15 (35 ply) rates Black as a little bit more than 3 pawns better after either 4...Nf6 5.Bxf7+ (1,749 games, White scores 42%) or 4...d6 5.Bxf7+ (39 games, White scores 33%).
In other words, if White castles before going "all in" on his Jerome attack, he reduces his position's evaluation and decreases his practical chances.
Likewise, if after 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 he first plays 4.Nc3,then against 4...Nf6 (3,231 games, White scores 39%) and 4...d6 (0 games) he still does worse than going directly for the sacrifices. Stockfish rates 4.Nc3 Nf6 5.Bxf7+ as a bit less than 3 pawns better for Black, and 4.Nc3 d6 5.Bxf7+ as more than 3 1/2 pawns better for Black.
All of this is actually a bit of slight-of-hand. The focus should not be on White. What actually is happening is that, given an extra move himself, Black strengthens his position before the Jerome sacrifices.
It is therefore no wonder that White, in delaying his attack (i.e. for his own King's safety), gives Black a chance to shore up his defenses.
The moral for White: Sacrifice away!